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Unformatted text preview: 65 .2405 TEST 3 April 24, 1996 Name 1. Please answer all questions, showing your work in detail and giving reasons where
appropriate. 2. This is an open book test. Collaboration with other students is NOT permitted. 3. Point allocations for each question are indicated. Plan your time accordingly. 4. In some questions it is speciﬁed that you solve the problem by hand. In all other questions you
are encouraged to use Maple whenever you think it will be helpful. 5. Be sure that you have all 8 test pages in addition to this cover sheet. NAME ___. ____..______________ 1. The eigenvalues and eigenvectors of a certain matrix A are 7'1: ’17 gm = T2 = —1/27 5(2) = (a) (6 points) Sketch a phase portrait for the system 2’ = Ax.
x7. (b) (6 points) Find the solution of the initial value problem NAME (C) (4 points) Sketch the graphs of 171 versus t and $2 versus t for the solution in part (d) (4 points) Classify the point x = 0 as to type (saddle point, node, spiral point, center)
and state whether it is asymptotically stable, stable, or unstable. NAME __ _________. _.______.. 2. (a) (10 points) Find by hand the eigenvalues and eigenvectors of the matrix B=(:§1‘)- (b) (10 points) The eigenvalues and eigenvectors of a certain matrix A are 3 3
r1 + 2’5 <2+z>’ T2 ’é 2—2“ Find the general solution, using real-valued functions, of the system I’ = Ax. NAME ____. ______________ __ 3. (a) (5 points) A phase portrait for a certain system 1’ = A3: is shown below.
. What can you say about the eigenvalues and eigenvectors of the matrix A? (b) (5 points) A phase portrait for a certain system 1" = A3: is shown below. Sketch
graphs of 1:1 versus t and $2 versus t that are consistent with this phase plot. Be
sure to identify which graph is 1:1 and which is 2:2. X1 Y (3,» NAME .______ (c) (5 points) One of the systems listed below has the phase portrait that is shown below.
Mark clearly the system having this phase portrait, and explain how you reached
your conclusion. ‘ NAME ____________________ 4. Consider the system x’ = 1 + 2y, y’ = l — 3x2 (a) (4 points) Find all of the equilibrium solutions (critical points). (b) (10 points) Choose a rectangle that includes all of the critical points. Construct a phase portrait for this syitem. Sketch your phase portrait below.
*2 (c) (6 points) From your phase portrait classify each critical point as to type (saddle point,
node, spiral point, center) and state whether it is asymptotically stable, stable, or
unstable. NAME _____________.. __ 5. The functions u (a: t) - sin m”: cos mm (1)
’f ’ _ 10 10 satisfy the wave equation
U11» = U“, 0 < :1: < 10, t> 0 the boundary conditions
u(0, t) = O, u(10, t) = 0, t > 0 (3) and the initial condition
ut(:r,0) = 0, 0 < :r < 10 (4) for all positive integers n. (a) (8 points) Suppose that we want to ﬁnd the function u(3:, t) that satisﬁes (2), (3), and
(4) and also the second initial condition u(:r, 0) = x(10 — z)2/125, 0 < a: <10 Write down the form of an inﬁnite series expression for u(a:, t), and provide an integral
formula for the coefﬁcients. (b) (7 points) Evaluate the coefﬁcients from part (a). NAME _._____________.____ (c) (5 points) Plot u(3,t) versus t for at least two periods, using a moderate number of
terms in the series. Sketch the graph of u(3, t) versus t below. u(3,t\ (d) (5 points) Plot u versus :2: for 0 g a: S 10 and for t = 7. Sketch the graph below. uhﬂ) lO ...
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